So off to home depot again today, buying saw blades. Turns out the saw originally came with the "fine" blades (out of fine, mediuim, coarse). The replacement set comes with 2 of each. Here's the difference between the fine and coarse blades:
This made a huge difference. Today I probably cut about 2 times as quickly as I did before, and felt comfortable taking out longer chunks at a time. A new record: 9.5 inches!
The following picture is a "timeline", or step-by step visual of how I shave down the sides after rough-cutting.
Starting from the right to left, you can see 3 definite sections. The first is the rough-cut, about 3-5 mm from the paper template. I start out by shaving the top part down to the paper (second section). I then gradually flatten this out till the whole portion is perpendicular to the grain, or kind of close enough...
I am of course a bit off each time, and this adds up. Some portions you can clearly see.